The topology of ridge systems

Topology is a branch of mathematics, concerned with the study of properties which are unchanged by continuous deformations. The geometrical concepts of ‘size’ or of precise ‘shape’ are not relevant to topology. If an object is distorted continuously without being torn apart or glued together, then its topology does not change, even though its precise geometrical configuration may become drastically altered. Thus, a solid sphere and a solid cube are topologically indistinguishable from one another whereas a solid sphere, a ring, and a hollow sphere are all topologically distinct. A topological classification of ridge patterns, therefore, will take no account of the normal geometrical relations such as ‘distance’ or ‘angle’. We may imagine the ridge patterns to be drawn on a rubber glove. The glove may be stretched or distorted in any way whatsoever (provided it is not torn or glued) and the topology of the patterns will not change. But it is dear that distance and angle measurements may be drastically altered by such deformations. In Fig. 1 , the circle (a) and the winding line ( b ) are topologically identical, whereas the open curve (c), although geometrically much more like a circle than ( b ) , is topologically distinct from (u,). I n many ways, a topological classification must be regarded as a rather crude one, since it ignores so much of the geometry. However, there are certain contexts in which topological ideas can prove very useful indeed. The subject has, in fact, been a very fruitful and actively studied one in modern mathematics. It also has applications in the physical sciences sometimes of a rather unexpected character. Perhaps it is less surprising that applications in the biological sciences are to be found in some areas. When a living thing grows, it is to be expected that considerable changes in size and shape may take place continuously, whereas changes in the topological structure might be imagined to occur only a t very special times in its development. In addition t o purely topological matters, there is another consideration that I feel may be important for classification schemes in biological contexts. This is the question of the stability of a classification. I n Fig. 2 is an illustration of what is an unstable situation, if we are concerned with the entire topological pattern of lines. It will be noticed that the two patterns are topologically quite distinct from one another, since the left-hand pattern contains several closed loops, whereas that on the right consists entirely of two open strands. However, only a very small locul distortion is needed in order to pass from the left-hand pattern to the right-hand one, namely a shunt upwards by one ridge of the portion to the right of the broken line. Any classification scheme which depends on tracing the connectivity of the entire pattern of lines is likely to be unstable in this sense. If the laying down of ridge patterns in a developing embryo is governed by purely local criteria, then the two patterns illustrated in Fig. 2 would have to be judged as being basically similar, even though they are topologically very different from one another. (If such differences in global pattern are to be found, for example, between identical twins even when the local pattern configurations are similar, then this could be viewed as lending support t o a hypothesis that the laying down of the ridges is indeed a Iocal process.)